DLMF:25.13.E2 (Q7738): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2aadec

Revision as of 13:50, 2 January 2020

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DLMF:25.13.E2
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    Statements

    Defining formula
    F ⁡ ( x , s ) = Γ ⁡ ( 1 - s ) ( 2 ⁢ π ) 1 - s ⁢ ( e π ⁢ i ⁢ ( 1 - s ) / 2 ⁢ ζ ⁡ ( 1 - s , x ) + e π ⁢ i ⁢ ( s - 1 ) / 2 ⁢ ζ ⁡ ( 1 - s , 1 - x ) ) , periodic-zeta 𝑥 𝑠 Euler-Gamma 1 𝑠 superscript 2 𝜋 1 𝑠 superscript 𝑒 𝜋 𝑖 1 𝑠 2 Hurwitz-zeta 1 𝑠 𝑥 superscript 𝑒 𝜋 𝑖 𝑠 1 2 Hurwitz-zeta 1 𝑠 1 𝑥 {\displaystyle{\displaystyle F\left(x,s\right)=\frac{\Gamma\left(1-s\right)}{(% 2\pi)^{1-s}}\*\left(e^{\pi i(1-s)/2}\zeta\left(1-s,x\right)+e^{\pi i(s-1)/2}% \zeta\left(1-s,1-x\right)\right),}}
    0 references
    DLMF id
    DLMF:25.13.E2
    0 references
    constraint
    ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}}
    0 references
    0 < x < 1 0 𝑥 1 {\displaystyle{\displaystyle 0<x<1}}
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    ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}}
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    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
    0 references
    Hurwitz zeta function
    Defining formula
    ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S11.E1.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
     
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